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arXiv:2111.09246

The Principle of equal Probabilities of Quantum States

M. Psimopoulos, Emilie Dafflon

Published Nov 17, 2021• 0 citations
arXivHTMLPDF

Abstract

The statistical problem of the distribution of $s$ quanta of equal energy $ε_0$ and total energy $E$ among $N$ distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by $p(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, s$ In this context, the special case ($N=4$, $s=4$) presented indicates that the alternative concept of most probable state is not valid for finite values of $s$ and $N$. In the present article we derive alternatively $p(κ)$ by distributing $s$ quanta over $N$ particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit $ε_{o} \rightarrow 0 $; $s \rightarrow \infty $; $s ε_{o} = E \sim$ fixed, where the energy of the particles becomes continuous, $p(κ)$ transforms to the Boltzmann law $P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\infty$ where $\langle ε\rangle = E/N$. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.

DOI
10.48550/arxiv.2111.09246

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The statistical problem of the distribution of sss quanta of equal energy ε0ε_0ε0​ and total energy EEE among NNN distinguishable particles is resolved using the conventional theory based on Boltzmann's principle of equal probabilities of configurations of particles distributed among energy levels and the concept of average state. In particular, the probability that a particle is in the \k{appa}-th energy level i.e. contains \k{appa} quanta, is given by p(κ)=(N+s−κ−2N−2)(N+s−1N−1)      ;      κ=0,1,2,⋯ ,sp(κ)=\displaystyle \frac{\displaystyle \binom{N+s-κ-2}{N-2}}{\displaystyle \binom{N+s-1}{N-1}} \;\;\; ; \;\;\; κ= 0, 1, 2, \cdots, sp(κ)=(N−1N+s−1​)(N−2N+s−κ−2​)​;κ=0,1,2,⋯,s In this context, the special case (N=4N=4N=4, s=4s=4s=4) presented indicates that the alternative concept of most probable state is not valid for finite values of sss and NNN. In the present article we derive alternatively p(κ)p(κ)p(κ) by distributing sss quanta over NNN particles and by introducing a new principle of equal probability of quantum states, where the quanta are indistinguishable in agreement with the Bose statistics. Therefore, the analysis of the two approaches presented in this paper highlights the equivalence of quantum theory with classical statistical mechanics for the present system. At the limit εo→0ε_{o} \rightarrow 0 εo​→0; s→∞s \rightarrow \infty s→∞; sεo=E∼s ε_{o} = E \simsεo​=E∼ fixed, where the energy of the particles becomes continuous, p(κ)p(κ)p(κ) transforms to the Boltzmann law P(ε)=1⟨ε⟩e−ε⟨ε⟩      ;      0≤ε<+∞P(ε) = \displaystyle \frac{1}{\langle ε\rangle}e^{-\fracε{\langle ε\rangle}} \;\;\; ; \;\;\; 0\leq ε< +\inftyP(ε)=⟨ε⟩1​e−⟨ε⟩ε​;0≤ε<+∞ where ⟨ε⟩=E/N\langle ε\rangle = E/N⟨ε⟩=E/N. Hence, the classical principle of equal a priori probabilities for the energy of the particles leading to the above law, is justified here by quantum mechanics.
One of the fundamental laws of classical statistical physics is the energy equipartition theorem which states that for each degree of freedom the average kinetic energy equals Ek=kBT/2E_k=k_B T/2Ek​=kB​T/2, where kBk_BkB​ is the Boltzmann constant and TTT is temperature of the system. Despite the fact that quantum mechanics has already been developed for more than 100 years still there is no quantum counterpart of this theorem. We attempt to fill this far-reaching gap and formulate the quantum law for equipartition of energy in the appealing form Ek=⟨Ek⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩, where Ek\mathcal E_kEk​ is thermal kinetic energy per one degree of freedom of the thermostat consisting of harmonic oscillators and ⟨...⟩\langle ...\rangle⟨...⟩ denotes averaging over frequencies ω\omegaω of those thermostat oscillators which contribute to EkE_kEk​ according to the probability distribution P(ω)\mathbb P(\omega)P(ω). We derive it for two paradigmatic and exactly solvable models of quantum open systems: a free Brownian particle and a harmonic oscillator. We formulate conditions for validity of the relation Ek=⟨Ek⟩E_k = \langle \mathcal E_k \rangleEk​=⟨Ek​⟩ for other quantum systems.
Statistical mechanics seems to be based on the idea of distributing a fixed amount of energy among a number of particles. (Usually this number is very large.) Alternatively, one may think of a single particle changing its state (e.g. momentum) during many time intervals. In both cases, one takes an average of the energy, but this means one needs weights for each single particle state. These weights seem to be based on the idea that each ‘vector’ of states has the same probability of occurrence. Thus, if one has N particles and an energy eN, one vector involves one particle having eN and the rest 0. Many vectors, however, may be created with N particles having energy near e. Thus, the weight for a single particle to have energy e is much higher than for it to have energy E. The weight is ultimately taken to be the Boltzmann factor. In quantum mechanics, it seems a very different assumption is made. Unlike classical physics where one follows a particle in space and time, knowing its acceleration, position and velocity at each x, in quantum mechanics, collisions with the potential seem to be granular, and the main assumption seems to be that a quantum particle may be in any momentum plane wave state at a given time. Like statistical mechanics, there is a set of weights, but one no longer uses the scenario of distributing energy among different states to determine these weights. Instead, the weights are obtained by noting that a quantum average at a point x is taken using fp exp(ipx)/ (Sum over fp exp(ipx)). The weights must ensure that: -1/2m (Sum over p p2/2m fp exp(ipx)) / (Sum over fp exp(ipx)) = E-V(x) at each x point. Thus, the plane wave behavior of quantum particles becomes a dominant factor. For the case of V(x)=0, the plane wave nature of the possible single particle momentum states leads to an overall (Sum over fp exp(ipx)) which has the appearance to two plane waves. This is very different from the classical statistical mechanical case. In this note, we attempt to examine the difference of these two statistical theories, and understand why in one case, the ground state of an oscillator, the two become the same.
Similarly to the derivation of the Gibbs-Boltzmann distribution for structureless indistinguishable particles, we consider multi-particle systems some of which are contained (or delimited) inside others (Problem 1), as well as systems of particles delimited within other particles, which, in turn, are delimited yet inside another kind of particles (Problem 2). Under the natural assumptions concerning the conservation laws, such as the conservation of the total number of particles, total energy, etc, the problem of the most probable energy distributions is studied in a combinatorial formulation, with the obtained distributions treated as externally observable. For Problem 1, the particle distributions over maximal and minimal energies are also established and shown to coincide with the ones found in the framework of the original combinatorial treatment. The results for Problems 1 and 2 can be interpreted as a sorting of lower-level particles based on their statistical mechanics behavior observed in various experiments. An effective Pauli principle arises in a non-contradictory manner in one-particle observations for the complete probabilistic Problem 1 in the combinatorial formulation as well as in the problem of distributions over maximal and minimal energies. Several of the calculated distributions describe particles having no negative energy states.
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
Abstract This work was trigerred by the earlier achivements of Yarman et al, aiming to bridge themordynamics and quantum mechanics, whence, Planck constant came to replace Boltzmann constant, and “average quantum level number” came to replace “temperature”. This evoked that the classical Maxwell energy probability distribution p(E) with respect to energy E of gas molecules might be taken care of, by the “energy probability distribution of the quantum levels” of a particle imprisoned in a given volume, assuming that in the case we have many particles, following Pauli exclusion principle, no pair of particles can sit at the same level. Thereby, the energy probability distribution of the quantum levels of a particle imprisoned in three dimensions, will be the subject of this essay. Such an outlook becomes interesting from several angles: i) It looks indeed very much like a classical Maxwellian distribution. ii) In the case we have as many free particles in the box as the number of levels depicted by the number of quantum levels in between the predetermined lower bound energy level and the upperbound energy level, all the while assuming that the Pauli principle holds, the distribution we disclose becomes the energy probability distribution of the ensemble of particles imprisoned in the given box. iii) It can even be guessed that, if elastic collisions between the free particles were allowed, and still assuming quantization and the Pauli principle, the outcome we disclose should be about the same as that of the energy probability distribution, molecules in a room would display in equilibrium. iv) The quantized energy being proportional to the sum of three squared integers associated with respectively, each of the spatial dimensions; the property we reveal certainly becomes remarkable from the point of view of mathematics of integer numbers. All the more, we further disclose that, to the probability distribution outlook remains the same, be this qualitatively for higher dimensions than 3.
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